Electric guitars are one of the defining elements of good ol' fashioned rock and roll music. From roots rock to death metal, you'll run into electric guitars pretty much every day in the live sound world. A few simple tips will put you on a solid path to rock guitar (mixing) greatness.
Listen to the Amp!
This sounds obvious, but you'd be surprised how many pro sound folks, myself included, will have already decided on the 'right' way to mic a guitar amp before the guitarist has even plugged in. Take a second and listen to the player's sound. Also, check with your ears to make sure that all the speakers in a cabinet are working. The owner of the cabinet is usually the last one to know when his speakers are torn, blown, or have fallen out and are rattling around inside the cab. (I've seen this.) You do not want to mic an empty hole, so check for yourself.
Choose your Weapons.
Your mic selection for electric guitar may come down to what's left over in your mic box. If you have options, though, don't be afraid to experiment. Dynamic microphones are usually preferred for guitar because of their ruggedness and their natural tendency to roll off low frequencies (the rumbly junk you don't need for guitars) and high frequencies (the really screechy scratchy stuff you're probably going to EQ out, anyway). However, certain particularly plucky condenser microphones are sometimes used to capture an edgier sound. If I have the option, I'll sometimes use both a condenser and a dynamic mic, and blend the sound to taste, but a desert island dynamic like a Shure SM57 is always a safe bet.
You'll occasionally see a third type of microphone—ribbon mics—used for guitars . . . but if you're using one of these, you either already know what you're doing or you've already ruined your microphone. Either way, no need for me to go down that avenue.
Location, Location, Location . . .
Think of microphone placement in terms of distance, angle, and position. The closer you place the mic to the speaker, the more direct sound you will get in relation to other sounds onstage. While some people in the studio world love the idea of ambient guitar mics, in live sound, close-micing is king. (Your 'ambient' guitar mic, after all, would also become an 'ambient' drum mic, an 'ambient' bass mic, and probably a feedback nightmare.)
Changing the angle of your microphone (in relation to the cabinet) can have a huge effect on the sound you capture. The specific change will depend on the pickup pattern of the particular microphone, but you'll often see an SM57 pointed at a 45 degree angle (called 'off-axis') instead of a straight-on 90 degrees ('on-axis'). Try it, you might like it.
Positioning the microphone is also a major ingredient in your guitar sound. We're talking inches. Micing the inside of a speaker cone will usually give you a brighter, edgier sound, while placing the mic toward the outside of the cone will tend to give more chunk and less sizzle. It isn't unusual to see professional touring bands (and local wannabes) mark their cabinets with bright tape to give the sound tech a helping hand in getting a good mic position.
Meanwhile, Back at the Board . . .
With live guitars, I try to keep the processing to a minimum. It's pretty easy to get carried away and EQ/compress/gate/whatever your guitar into oblivion. I'll generally use a high-pass filter to cut some rumble, and maybe low- and high-shelf filters to cut a little mud and screech. Other than that, I'll usually take a pretty laissez-fare approach unless there's a specific problem I want to address.
One of the most common challenges you'll encounter, especially with guitar-centric bands, is that the vocal and the guitar occupy a lot of the same upper-midrange space. In this case, I might find a range where the vocal really jumps out (say, 3 khz) and boost it a little. Then, I'll dip the same range in the guitar, while maybe boosting the guitar/dipping the vocal at 2 or 4 kHz. It's subtle, but this complimentary EQ helps to give each element its own space. In live sound, it's easy to make things loud, but the real trick is to do so while maintaining clarity.
Know Thy Purpose . . .
In rock music, electric guitars serve many different functions, from light chords to blistering leads to sounds that sound like anything but a guitar. Take a listen, pick a mic, put it in a spot, and massage the final sound into the mix. After that, it's just rock and roll.